I am looking for a method / Linux command to get the up-time of the open-ssh server and Nginx server running on my Ubuntu system. I want to know the up-time for calculating some internal statistics. I am looking for something similar to mysqladmin status command.

That file is updated when the daemon starts/restarts. So you can calculate based on that.

For nginx you can install the status module (http://wiki.nginx.org/HttpStubStatusModule) and then use that to get some information, but alas uptime doesn't look to be there. Could probably use the same idea of looking at the nginx pid files, but that will just tell you since the last restart etc. and not 100% indicative of uptime.